Wavefunctions Pauli exclusion principle

Introduction of the half-integral spin of the electrons (values h/2 and —fe/2) alters the above discussion only in that a spin coordinate must now be added to the wavefunctions which would then have both space and spin components. This creates four vectors (three space and one spin component). Application of the Pauli exclusion principle, which states that all wavefunctions must be antisymmetric in space and spin coordinates for all pairs of electrons, again results in the T-state being of lower energy [equations (9) and (10)]. 

The properties of the Slater determinant demonstrate immediately the Pauli exclusion principle, as usually taught. It reads No two electrons can have all four quantum numbers equal, that is to say that they cannot occupy the same quantum state. It is the direct result of the more general argument that the wavefunction must be antisymmetric under the permutation of any pair of (identical and indistinguishable) electrons. 

For over a decade, the topological analysis of the ELF has been extensively used for the analysis of chemical bonding and chemical reactivity. Indeed, the Lewis pair concept can be interpreted using the Pauli Exclusion Principle which introduces an effective repulsion between same spin electrons in the wavefunction. Consequently, bonds and lone pairs correspond to area of space where the electron density generated by valence electrons is associated to a weak Pauli repulsion. Such a property was noticed by Becke and Edgecombe [28] who proposed an expression of ELF based on the laplacian of conditional probability of finding one electron of spin a at t2, knowing that another reference same spin electron is present at ri. Such a function... 

The antisymmetry of many-electron spin-orbital products places constraints on any acceptable model wavefunction, which give rise to important physical consequences. For example, it is antisymmetry that makes a function of the form I Isa Isa I vanish (thereby enforcing the Pauli exclusion principle) while I lsa2sa I does not vanish, except at points ri and 1 2 where ls(ri) = 2s(r2), and hence is acceptable. The Pauli principle is embodied in the fact that if any two or more columns (or rows) of a determinant are identical, the determinant vanishes. Antisymmetry also enforces indistinguishability of the electrons in that Ilsals(32sa2sp I =... 

Following the Pauli exclusion principle we must antisymmetrize the wavefunctions when we include the spins of the two electrons. The wavefunction is given by a product of spatial and spin wave functions, i.e. 

As pointed out earlier, the physical basis of this repulsion is the increase in kinetic energy of the electrons due to the Pauli exclusion principle, which is most easily seen from the large gradients induced in the wavefunctions by the orthogonality requirement. 

Slater determinants enforce the Pauli exclusion principle, which forbids any two electrons in a system to have all quantum numbers the same. This is readily seen for an atom if the three quantum numbers n, l and mm of ij/(x, y, z) (Section 4.2.6) and the spin quantum number ms of a or /i were all the same for any electron, two rows (or columns, in the alternative formulation) would be identical and the determinant, hence the wavefunction, would vanish (Section 4.3.3). 

The Hartree-Fock equations (5.47) (in matrix form Eqs. 5.44 and 5.46) are pseudoeigenvalue equations asserting that the Fock operator F acts on a wavefunction i//, to generate an energy value ,-, times i/q. Pseudoeigenvalue because, as stated above, in a true eigenvalue equation the operator is not dependent on the function on which it acts in the Hartree-Fock equations F depends on i// because (Eq. 5.36) the operator contains J and K, which in turn depend (Eqs. 5.29 and 5.30) on i//. Each of the equations in the set (5.47) is for a single electron ( electron 1 is indicated, but any ordinal number could be used), so the Hartree-Fock operator F is a one-electron operator, and each spatial molecular orbital i// is a one-electron function (of the coordinates of the electron). Two electrons can be placed in a spatial orbital because the, full description of each of these electrons requires a spin function 7 or jl (Section 5.2.3.1) and each electron moves in a different spin orbital. The result is that the two electrons in the spatial orbital i// do not have all four quantum numbers the same (for an atomic Is orbital, for example, one electron has quantum numbers n= 1, / = 0, m = 0 and s = 1/2, while the other has n= l,l = 0,m = 0 and s = —1/2), and so the Pauli exclusion principle is not violated. 

To construct the HF determinant we used only occupied MOs four electrons require only two spatial component MOs, pi and i//2, and for each of these there are two spin orbitals, created by multiplying ijj by one of the spin functions a or jl the resulting four spin orbitals (i/qa, pi/31p2x, i//2/ ) are used four times, once with each electron. The determinant the HF wavefunction, thus consists of the four lowest-energy spin orbitals it is the simplest representation of the total wavefunction that is antisymmetric and satisfies the Pauli exclusion principle (Section 5.2.2), but as we shall see it is not a complete representation of the total wavefunction. 

Such a function vanishes because any determinant with two identical rows or columns vanishes. In other words, any system having both electrons in the Is orbital with a spin cannot exist. Now we see there is an alternative way of saying the Pauli Exclusion Principle A wavefunction for a system with two or more electrons must be antisymmetric with respect to the interchange of labels of any two electrons. 

Fock recognized that the separable wavefunction employed by Hartree (Eq. (1.6)) does not satisfy the Pauli exclusion principle. Instead, Fock suggested using the Slater determinant... 

This restriction is not demanded. It is a simple way to satisfy the Pauli exclusion principle, but it is not the only means for doing so. In an unrestricted wavefunction, the spin-up electron and its spin-down partner do not have the same spatial description. The Hartree-Fock-Roothaan procedure is slightly modified to handle this case by creating a set of equations for the a electrons and another set for the p electrons, and then an algorithm similar to that described above is implemented. 

A is the antisymmetrizer, ensuring that the wavefunction changes sign on interchange of two electrons (and thus the wavefunction obeys the Pauli exclusion principle), and 0(S) is a spin projection operator " that ensures that the wavefunction remains an eigenfunction of the spin-squared operator,... 

The determinant 4/, the HF wavefunction, thus consists ofthe four lowest-energy spin orbitals it is the simplest representation of the total wavefunction that is antisymmetric and satisfies the Pauli exclusion principle (section 5.2.2), but as we shall see it is not a complete representation of the total wavefunction. 

One way of phrasing the Pauli exclusion principle is the requirement that the electronic state wavefunction 4> be antisymmetric with respect to the interchange of any two electron coordinates. For example,... 

This additive density approximation does not correspond to the antisymmetrized product of the ionic wavefunctions which give the densities Pj,- however, the resulting electron density does correspond to some antisymmetrized (although unknown) wavefunction, and thus does not violate the Pauli exclusion principle [15]. If there were full variability... 

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